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Abstract: The dodecahedron is a beautiful shape made of 12 regular pentagons. It doesn’t occur in nature; it was invented by the Pythagoreans, and we first read of it in a text written by Plato. We shall see some of its many amazing properties: its relation to the Golden Ratio, its rotational symmetries - and best of all, how to use it to create a regular solid in 4 dimensions! Poincaré exploited this to invent a 3-dimensional space that disproved a conjecture he made. This led him to an improved version of his conjecture, which was recently proved by the reclusive Russian mathematician Grigori Perelman - who now stands to win a million dollars.

(By the way, when I say that the Pythagoreans invented the dodecahedron, I’m not claiming nobody else invented it first! According to Atiyah and Sutcliffe, these blocks found in Scotland date to around 2000 BC:

Should we count them as Platonic solids even if they’re rounded? That’s too tough a puzzle for me. Still less do I want to get into the question of whether the Platonic solids were invented or discovered!)

Posted at November 12, 2006 4:40 AM UTC

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Re: Tales of the Dodecahedron

Re: Tales of the Dodecahedron

I’m glad you like those slides! I was trying out a new method: using web pages, instead of making PDF files with LaTeX. For a talk with lots of pictures and movies, and not many equations, it seems to work well.

Re: Tales of the Dodecahedron

You should count them as Platonic solids if they have the same symmetry group as (the polygonal) Platonic solids, right?

Well, no - since then we’d have to count the cube and octahedron as the same, since they have the same symmetry group. The dodecahedron and icosahedron also have the same symmetry group.

So, there’s more to the essence of a regular polytope than its symmetry group. The question is: what?

In trying to answer this, modern mathematicians have decided that regular polytopes drawn on a sphere are more fundamental than those with planar faces. If we include these, the classification becomes more systematic! In addition to the usual Platonic solids, we get the hosohedra with nn bigons as faces, like this:

Including hosohedra and dihedra may seem weird at first, but it basically amounts to not discriminating against the number 2: we allow 2-sided polygons and allow 2 polygons to meet at an edge. This is naturally suggested by the Schläfli symbols for regular polytopes - those funny little symbols next to the pictures above, which are part of a very nice mathematical theory.

So, it seems the early inhabitants of the British isles were in some ways further along than the Greeks in understanding the deep inner meaning of Platonic solids. I know McKay, and he likes this - national pride I guess.

However, I don’t know of any stone spheres that look like hosohedra or dihedra. The realization that 2 is a number just like 3, 4, 5,… came surprisingly late in the history of mathematics - perhaps because many languages, including early Celtic languages, have singular, plural and dual forms of nouns and pronouns.

I know your remark had a smiley under it, but I couldn’t resist taking it seriously, since it leads to some interesting issues.

Re: A counterexample to the claim that…

By the way, I think those models from Scotland are the most amazing thing I’ve seen in a long time. Is there any chance they’re a hoax? I’m astounded that someone in the culture of Scotland around 2000BC had the time, inclination and insight to make those models. This is the work of someone with a profound understanding of geometry. Could one person have come up with these from nothing, or was there a now lost intellectual culture so that these models are the result of the thought of many people over an extended period? Either way, it’s surprising for what was a neolithic culture. Given the popularity of stone circles in that part of the world, maybe there was a well developed culture of geometry. I wonder what other evidence of it exists.

Those Scots had balls

Dan Piponi writes:

By the way, I think those models from Scotland are the most amazing thing I’ve seen in a long time.

Yeah, they’re cool.

Is there any chance they’re a hoax?

According to Alison Roberts, collection manager of the Ashmolean at Oxford, such carved balls have been found at late Neolithic and early Bronze Age sites throughout Scotland, with a few in North England and Ireland too. If they’re all hoaxes, that would have taken some work!

Could one person have come up with these from nothing, or was there a now lost intellectual culture so that these models are the result of the thought of many people over an extended period?

If they’re all over Scotland, they’re probably the work of many people.

Either way, it’s surprising for what was a neolithic culture. Given the popularity of stone circles in that part of the world, maybe there was a well developed culture of geometry.

Maybe! For what it’s worth, stone balls of this sort date from the Late Neolithic to the Early Bronze Age: 2500 BC to 1500 BC.

By comparison, the megaliths at Stonehenge date back to 2500-2100 BC, with some bits going back to 3100 BC, and some nearby Mesolithic postholes all the way back to 8000 BC. Building Stonehenge took some serious social organization, and maybe some serious interest in geometry and astronomy too. I’ve read some cool theories about that, but they’re really controversial.

So, I guess we only see a trace here and there of what could have been some fairly sophisticated cultures. Too bad we can’t travel back in time and learn more…

I’ve attached the early papers on the first synthesis of dodecahedrane, C20H20, which was apparently pursued for about 20 years before total success. If you look at the papers, keep in mind that it is common practice in organic chemistry to leave out bonds to hydrogen atoms in the pictures. A vertex from which three lines emanate is understood to mean a carbon atom bonded to a hydrogen atom that is not shown.

One of my favorite molecules is alpha-boron, B12. Natural elemental boron has more than one allotrope (chemically distinct forms of an element, like dioxygen, O2, and ozone, O3), one of which is B12. The boron atoms are arranged so that each sits at a vertex of a regular icosahedron. This was known for years before C60 was discovered, but it never got as much press.

The chemical syntheses of dodecahedrane look very pretty. For example, here are some reactions from the second paper:

Experts on higher category theory will recognize that, if we ignore the methyl groups and other radicals, this is precisely the proof of the “dodecahedral identity” satisfied by the pentagonator in any monoidal 2-category. Here the proof itself has been categorified, with the steps of the proof corresponding to the arrows between diagrams. A truly amazing example of the unity of math and physics!

Re: Tales of the Dodecahedron

Experts on higher category theory will recognize that, if we ignore the methyl groups and other radicals, this is precisely the proof of the “dodecahedral identity” satisfied by the pentagonator in any monoidal 2-category.

(Just kidding.)

Of course, there is an identity satisfied by that pentagonator (and more generally by the pentagonator in any 3-category), but it only has 6 pentagons in it (as well as 3 quadrilaterals for the naturality of the associator). So it’s a (nonregular) ennahedron, which I suppose one calls simply ‘associahedron’.

I think that Aaron Lauda (and Eugenia Cheng) had a diagram of this online that you could print, cut out, and put together, but maybe it was a diagram of something else. In any case, I can’t find it now, but it’s on page 14 here.

I hadn’t expected Scottish antiquaries to have gone online… I imagined them being a bit dusty and old-fashioned.

Executive summary: 387 carved stone balls have been found in Scotland, dating from the Late Neolithic to Early Bronze Age, with a wide variety of interesting geometric patterns carved on them. You can see lots of pictures of them in this article. Here are two of the maps showing where various types of balls have been found:

Re: Tales of the Dodecahedron

007 vertices of a 6-simplex; Re: Tales of the Dodecahedron

Greater doubt is cast on neolithic Scots as classifiers of, and carvers in stone, of the 6 platonic hypersolids. And the alleged stone-aged small stellated dodecahedron is not a premature Kepler-Poinsot solid, but merely the spiky ball of a mace from a William Wallace statue intended for Mel Gibson’s beachhouse in Malibu. Scottish Nationalist Party stalwart Sir Sean Connery refused to speak to this reporter.

I was a teenaged Polyhedron; Re: Tales of the Dodecahedron

Thursday 16 April 2009 I spent an hour at Abraham Lincoln High School in Los Angeles having 33 9th-12th graders build the Platonic solids with paper, scissors, and glue. They particularly liked the regular dodecahedron.

I’d have the students then come to the white board with their constructs and fill in a line each of a table on the number of vertices, edges, and faces. Did so also with my models of pentagonal pyramid and triangular prism. Then, Friday 17 April 2009, had those who hadn’t finished the Platonic solids build some Archimedean solids. I’ve just finished doing statistics on a 10-question survey, which 29 of the 33 students completed.

Previously, I’d done this at a different high school in Los Angeles. To control free variables, both had predominantly Latino students, and both built shapes and filled out surveys on Friday.

I have nice powerpoint graphics of the survey results from the 1st school. Will do so this weekend for this 2nd school. Will give my AP Statistics Class a chance to crunch the numbers more deeply to verify what appear to be significant differences between boys and girls in this project, and to explore the correlations with survey questions of aesthetics, the amount of fun they had, the changes in their feeliongs about Geometry, and their belief that such work increases intelligence. No students yet have self discovered Euler’s polyhedral formula. So I figure that to take a bigger sample, or more advanced students, or a longer sequence of days for this solid geometry unit.